Why do people have to be nude during a massage?

Not all types of massage require the recipient to remove their clothing.

For those types of massage which do require the removal of clothing, it is expected that client's arrive to their massage CLEAN and LEAVE ON A PAIR OF UNDERPANTS so that professional draping with a sheet can be performed during the massage.

For the type of massage most people are familiar with, Swedish massage, removal of the clothing allows the therapist to more easily feel tense muscles, see the skin and body for signs of distress, and most clients enjoy the sensation of oil or lotion being applied. Without clothing, the therapist can use long sweeping strokes with more pressure, work more easily on areas like beneath the shoulder blades, and have an easier time manipulating tissues.